I've read in several places that Altered Carbon just takes place on Earth, several hundred years from now. But nearly every external shot at night seems to show two moons in the sky, and neither one is in the same position or relative size as the actual Earth Moon. So where does the main story of Kovachs investigating the murder take place?
It takes place in Bay City, Earth
(former San Francisco)
As stated, in the first episode Kovacs asks on which planet he is.
[INHALES DEEPLY] Where am I?
Alcatraz Prison. Bay City.
What planet, genius?
Both Alcatraz and the Golden Gate Bridge (as well as the city's name) indicate that Bay City was built on what was known as San Francisco.
As noted in the comments, all scenes which included several moons are located on Harlan's World and are mostly flashbacks. I have watched the whole series and I am sure that every location is presented consistently.